The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

"Seven-Color Torus Series in Bead-Crochet: Bracelet 2," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html

"Seven-Color Torus Series in Bead-Crochet: Bracelet 1," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html

"Seven-Color Torus Series in Bead-Crochet: Bracelet 3," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html

"Seven-Color Torus Series in Bead-Crochet: Bracelet 4," by Sophie Sommer (Colgate University, Hamilton, NY; Susan Goldstine (St. Mary’s College of Maryland, St. Mary’s City); Ellie Baker (Computer scientist/Artist, Lexington, MA) Bead-crochet (glass beads, thread) , "11.25” x "11.25”, 2008-2009. One of a series of “map-coloring” bead-crochet bracelets. The first three are examples of maps on the torus where each of seven “countries” shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan’s seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process." --- http://faculty.smcm.edu/sgoldstine/torus7.html

"Ribbons of Rhythm," by Paul Stacy (Landscape Architect, Sydney, Australia)Giclee digital print, 22" x 14" , 2009. Ribbons of Rhythm (foreground image with detail behind) is an exploration of the aesthetic qualities of Penrose tiling. David Austin in "Penrose Tilings Tied up in Ribbons", describes the ribbons thus: "Opposite sides of a rhomb are parallel to one another. Therefore, if we begin with a rhomb and a pair of opposite sides, we may form a "ribbon" by adding the rhombs attached to that pair of opposite sides and then continuing outward". The print reveals only a single family of parallel ribbons in one orientation, however there are another four orientations associated with a five-fold Penrose tiling. "Without a programming or mathematical background, I explore my interest in Penrose tiles by hand building patterns in Corel Draw and experimenting with colors, shapes, welding, contouring and other functions that allow me to explore the aesthetic realm of Penrose tiling, which continues to hold my interest particularly as long range positional order and beauty are revealed." --- Paul Stacy (Landscape Architect, Sydney, Australia) www.pdstacy.com

"Experiment in Shading," by Norton Starr (Amherst College, Amherst, MA)(Pressurized) ball point pen on paper, 14.25” wide by 15.25” high, 1973. This was drawn by computer-controlled pen on a CalComp Drum plotter at the University of Waterloo. It consists of several hundred concentric star images, with their “radius” varied sinusoidally so as to create the shadow effects of darker and lighter regions. The end result is like an unrealistically precise charcoal drawing. "As I grew up, my freehand drawing often involved families of parallel lines and curves suggesting shading effects. In 1972 I recognized that with the aid of a computer driven plotter I could obtain pictures essentially impossible by other means. Although I produced a number of drawings of different kinds, I spent a fair amount of time and effort trying to achieve shading effects by drawing lines and curves variably spaced from one another. The computer afforded a degree of control that made possible my use math functions to provide desired transitions between dark and light regions. 'Experiment in Shading' is one consequence of that initiative." --- Norton Starr (Amherst College, Amherst, MA) http://www3.amherst.edu/~nstarr/

"Tying and untying," by Victor Stipsic (Washington DC), Marko Vujic (Washington DC), Radmila Sazdanovic Movie Clip, 2009. "Tying and untying" is a short movie addressing one of the principal questions in knot theory-unknotting and distinguishing knots. More precisely, we illustrate John H. Conway’s classification of knots into knot and link families. Mathematical ideas permeate vivid animations and music creating visual-acoustic symphony. --- Victor Stipsic (Washington DC), Marko Vujic (Washington DC), Radmila Sazdanovic http://www.youtube.com/watch?v=Bz_A6nhrZMw

"Perspective Sphere," by Dick A. Termes (Artist, Spearfish, SD)Acrylics on Polyethylene sphere, 10" diameter sphere, 2008. The "Perspective Sphere" is the story of perspective. It shows a 360 degrees in all directions cityscape which is organized with a six point perspective system. This means, every line drawn on the sphere makes a greater circle and every cubical building projects to all six vanishing points which are equally spread around the sphere. This piece also shows examples within the spherical painting of a one point perspective, a two, three, four, five and six point perspective. All are sectioned off with the use of color. "I paint what are called Termespheres. These are inside out total worlds that are painted on spheres that hang and rotate from ceiling motors. These spherical paintings show you up, down and all around environments. Some are interiors of famous buildings and some are outside scene, some are also geometric studies and others are subconscious worlds that I imagine are around me. Most of these explore a six point perspective system which I find to be more true than any of the other systems of perspective. This is my 40th year of exploring work like this on the spherical canvas. I feel this exploration has opened up a new way to see the world and its geometry tells me it is more than just art. I think it connects with math very well." --- Dick A. Termes (Artist, Spearfish, SD) www.termespheres.com

"Icosahedron #1," by Briony Thomas (School of Design, University of Leeds, UK)Laser-cut acrylic, 6.5" x 5.5" x 6", 2007. The successful application of a pattern to repeat across the faces of a polyhedron is determined by the pattern's underlying lattice structure and its inherent symmetry operations. Only pattern classes containing six-fold rotation are applicable to patterning icosahedron. Icosahedron #1 exhibits a p6 pattern cut from the faces of the solid. Centres of six-fold rotation in the pattern become axes of five-fold rotation at each vertex and all other rotational symmetries are preserved. "As a designer, with a background in textiles, I am fascinated by the fundamental concept of symmetry and its application in the creation of patterns. This recent work explores the possibilities of patterns repeating in three-dimensions, around the faces of mathematical solids." --- Briony Thomas (School of Design, University of Leeds, UK)

"Solar System," by Eve Torrence (Randolph-Macon College, Ashland, VA)Watercolor on paper, 8.5'' x 8.5'' x 8.5", 2009. This polyhedron is comprised of ten tetrahedra. Two mirror-image compounds of five tetrahedra are merged to form the solid. When the polyhedron is rendered in a single color it is difficult to distinguish the individual tetrahedra, in part because some pairs of faces are coplanar. To help the viewer resolve this visual puzzle, the ten tetrahedra have been painted with distinct patterns and colors, which are suggestive of the Sun and the nine planets. The overall star-like quality of the polyhedron, and the tight entwining of the tetrahedral "planets", is evocative of our solar system. "I love the symmetric beauty of polyhedra and enjoy using paper to create models to study. Through the process of creating a model I am able to truly understand its structure. My own curiosity about the underlying structure of this compound of ten tetrahedra led me to make a multicolored model. I was inspired by the 2009 exhibit 'Images of the Universe from Antiquity to the Telescope' celebrating the 400th anniversary of Galileo's discovery of our moon's craters. This model pays homage to Renaissance depictions of the solar system that used various polyhedra to model the celestial bodies." --- Eve Torrence (Randolph-Macon College, Ashland, VA)

"Fermat Point," by Suman Vaze (King George V School, Hong Kong)Acrylic on canvas, 20” x 24”, 2008. The Fermat Point of a triangle is the point of least total distance from the vertices of a given triangle. The painting depicts that the Fermat Point of a triangle can be obtained by constructing equilateral triangles on each side and then joining the vertices of the original triangle and the equilateral triangles. It also shows that circles with the sides of the triangle as chords also intersect at the Fermat Point. "I seek to depict interesting mathematical truths, curiosities and puzzles in simple, visually descriptive ways. Mathematical amusements inspire the color and form in my paintings, and I try to strike a balance between the simplicity of the concepts and their depiction in art. The logic and balance of the discipline is beautiful, and I like art that both stills and stimulates the mind--these are the qualities I strive to capture in my work." --- Suman Vaze (King George V School, Hong Kong) http://vazeart.googlepages.com

"You and me, and an army of monkeys," by Samuel Verbiese (Artist, Overijse, Belgium)Computer image printout glued on cartonboard panel (vZome program by Scott Vorthmann), 17x12 inches (A3, 420x297 mm), 2009 (2005 for the concept of the Golden Pyramid, shown at Bridges London). My 'Golden Pyramid' is a truss that can project (when viewed from underneath, at a precise, quite near point, orthogonally to the back golden triangular face) into the K5 graph (pentagram inscribed in a pentagon) with remarkable proportions (two equilateral and two golden triangles on a golden rectangular base featuring its two diagonals). The spacecraft looking model shown has its struts built here, kind of fractally, from 463 overlapping tiny golden pyramids (that can be 3D-copied). Thanks to a most welcome serendipity, the chosen view angle gives the attentive viewer the needed substance to the title of this ludical work: on the almost vertical strut in the center of the image where it crosses two other struts, two bearded 'wise' men appear -sorry for the ladies : the serendipity unfortunately didn't help them here ! - and on the remaining part of that strut, a series of monkey faces... "Besides expressionistic painting and sculpting of the figure and portrait, I am recurrently drawn into geometric projects, probably by previous life." --- Samuel Verbiese (Artist, Overijse, Belgium)

"RUBIK's new clothes," by Anna Viragvolgyi (Mathematician, Budapest, Hungary)Stirol cube with plastic foil, 2,5"x2,5"x2,5", 2009. Example of extending pattern of "48 different squares" over the surface of RUBIK's 4x4x4. Each square of the set appears twice on the 96 tiles of the cube. There are various symmetries on the sides of the cube and between the sides also. So there is more than one coherent and continuous arrangement. "I deal with diagonally striped, coloured squares. [These squares assign a restricted de Bruijn sequence S(k,n). There are [k(k-1)^(n-1)]/2 distinct squares, where k is the number of colours, n is the number of stripes.] Last year I studied the geometric shape of arrangements of the squares [in case of k=3, n=6, S(3,6)=48] with coherent pattern on the plane. Presently my aim is filling surfaces of solid figures with these squares. Here is one of them." --- Anna Viragvolgyi (Mathematician, Budapest, Hungary) viragvolgyi.anna@gmail.com

"The Three Gates," by Benjamin Wells (University of San Francisco, CA)Inkjet on transparencies, mounted on acrylic layers in an acrylic frame hung by a laced beadchain, 11" x 9", hung by a beadchain, plus 11"x7" explanatory placard 2009 (updated from 2002). Symbols from logic map a classical aphorism about watching one's tongue into a visually recursive statement. In addition, the colors of red and green play on the binary nature of electronic gates. An accompanying placard gives the aphorism and lists the symbols used. "The art offered is a melding of symbolism from science and mysticism. The flexibility of computer-aided design and execution supports this blend of ancient and modern expression. I used to think in threes because my name ends in III. (For more about small numbers, see Michael Schneider’s 'Constructing the Universe.') Although I am now partial to 8, 17, 36, 1111, and 10^10, I wholly support only 1 alone. But three things can start a sequence, give a contrast or equivalence, or triangulate. Here they pose visual riddle. Math is fun, and art can help make that clear. When it can also take a supportive, spiritual, inspirational, cooperative color, then it is a harbinger of a new humanity. I hope to make art that way." --- Benjamin Wells (University of San Francisco, CA)